hello I need help
I spent 2 hours without getting the right solution
=
[1-[1-P(X1<=y1)[1-P(X2<=y1)][P(X1<=y2)P(X2<=y2)]
but the book gives a different aswer from what I got
I think I cannot assume that Y1 and Y2 are independant right?

Thank you

January 1st 2010, 02:47 PM

matheagle

This is the joint distribution of two order statistics from a sample of two.
You can find that in many books.
I think it's in Wackerly, I used to use Hogg and Craig.
since you only have two observations...

There are 2! ways this can happen.
Either Y1=X1 or Y1=X2.
The reason it's still a valid density is that we now have a restriction on the space.
The order statistics are dependent.

January 2nd 2010, 12:23 AM

Moo

Hello,

@ qwerty321 :
No they're not independent, so your working is unfortunately wrong.
First consider (since it's continuous random variables, it doesn't matter whether is or >)

If the min is superior to a y1 and the max is inferior to y2, it simply means that

So

And then you can use the independence between X1 and X2 to calculate this probability.

Then

Hence

the missing part is

can you finish it ?

You have to visualize that if the min of two values is > to something, then the two values are > to something. If the max of two values is < to something, then the two values are < to something.

January 2nd 2010, 12:43 AM

qwerty321

matheagle i cannot understand how you gor your formula
Moo I do not understand :

first why you took this case:
"
First consider (since it's continuous random variables, it doesn't matter whether is or >)"

second:
then how did you pass from

"

to this "

and for the missing part i guess you mean i could use the independance thing for X1 and X2

thank you

January 2nd 2010, 05:52 AM

Moo

Quote:

Originally Posted by qwerty321

Moo I do not understand :

first why you took this case:
"
First consider (since it's continuous random variables, it doesn't matter whether is or >)"

Because it's easy to deal with min if it's > to something, and to deal with max if it's < to something (see my last sentence in the first post).

Quote:

second:
then how did you pass from

"

to this "

erm... a=b+c ===> b=a-c ?

Quote:

and for the missing part i guess you mean i could use the independance thing for X1 and X2